What is ln(1/θ*[1-1/θ ]^{x-1}) ?

The solution to ln(1/θ*[1-1/θ ]^{x-1}) is ln(-1/θ+1)(x-1)-ln(θ)

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simplify ln(1/θ*[1-1/θ ]^{x-1})

simplify

ln (\frac{1}{θ} \cdot [1 - \frac{1}{θ}]^{x - 1})

ln (- \frac{1}{θ} + 1) (x - 1) - ln (θ)

Solution steps

ln (\frac{1}{θ} [1 - \frac{1}{θ}]^{x - 1})

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

ln (\frac{1}{θ} [1 - \frac{1}{θ}]^{x - 1}) = ln (\frac{1}{θ}) + ln ([1 - \frac{1}{θ}]^{x - 1})

= ln (\frac{1}{θ}) + ln ([- \frac{1}{θ} + 1]^{x - 1})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ([1 - \frac{1}{θ}]^{x - 1}) = (x - 1) ln (1 - \frac{1}{θ})

= ln (\frac{1}{θ}) + (x - 1) ln (1 - \frac{1}{θ})

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{1}{θ}) = ln (1) - ln (θ)

= ln (1) - ln (θ) + ln (- \frac{1}{θ} + 1) (x - 1)

Apply log rule:

lo g_{a} (1) = 0

= 0 - ln (θ) + ln (- \frac{1}{θ} + 1) (x - 1)

0 - ln (θ) + (x - 1) ln (1 - \frac{1}{θ}) = - ln (θ) + (x - 1) ln (1 - \frac{1}{θ})

= ln (- \frac{1}{θ} + 1) (x - 1) - ln (θ)

ln (3 b \frac{b + 1}{b - 1})

5 ln (x) - \frac{1}{4} ln (y)

lo g_{4} (4 x^{2})

lo g_{4} (\frac{64}{x + 5})

3 lo g_{3} (x) - lo g_{3} (\frac{1}{y})

What is ln(1/θ*[1-1/θ ]^{x-1}) ?
The solution to ln(1/θ*[1-1/θ ]^{x-1}) is ln(-1/θ+1)(x-1)-ln(θ)